No. For example, in the category of topological abelian groups, Z is far from injective. Nonetheless, if you say that a group is Z-compact when it is an equalizer of two maps between powers of Z, then an equalizer of two maps between Z-compact abelian groups is again Z-compact. The proof is not direct. As it happens, I am talking on this in our seminar tomorrow. Even though the reals are not injective in hausdorff spaces, a space is realcompact iff it is a closed subspace of a power of R, which turns out to be equivalent to being an equalizer of two maps between powers of R (that is a cokernel pair of such a closed inclusion has enough real-valued functions to separate points) and it is clear that a closed subspace of a realcompact space is again realcompact. Same thing for N-compact. In fact, for every example I have looked at sufficiently closely. Michael On Mon, 12 May 2008, Eduardo Dubuc wrote:
Consider the dual finitary question: In universal algebra in order to show that finitely presented objects are closed under coequalizers it is essential that a amorphism of finitely presented objects lift to a morphism between the free. Is this the only way to prove it ? :
" but when I look at examples, it has turned out to be true for other reasons."
greetings e.d.
In March I asked a question on adjoints, to which I have received no correct response. Rather than ask it again, I will pose what seems to be a simpler and maybe more manageable question. Suppose C is a complete category and E is an object. Form the full subcategory of C whose objects are equalizers of two arrows between powers of E. Is that category closed in C under equalizers? (Not, to be clear, the somewhat different question whether it is internally complete.)
In that form, it seems almost impossible to believe that it is, but it is surprisingly hard to find an example. When E is injective, the result is relatively easy, but when I look at examples, it has turned out to be true for other reasons. Probably there is someone out there who already knows an example.
Michael